BIOSIGNAL 2010
Multi-resolution Linear Model Comparison for Detection of
Dicrotic Notch and Peak in Blood Volume Pulse Signals
Rudolf B. Blažek 1, Chien-Cheng Lee 2
Communication Research Center, Yuan Ze University, Taiwan
2
Department of Communications Engineering, Yuan Ze University, Taiwan
[email protected]
1
This paper proposes a novel approach for detection of dicrotic notches and peaks from blood
volume pulse signals that are measured by non-invasive photoplethysmography sensors. Dicrotic notches and peaks represent the closure of the aortic valve and subsequent retrograde
blood flow. Their location can be used to calculate systolic time intervals and monitor cardiac function. Photoplethysmography sensors are usually placed on the patients’ fingers,
ears, or toes. The signals are therefore often distorted by poor periphery perfusion or motion
artifacts. The proposed method is robust and overcomes these artifacts even in cases where
the dicrotic notch and peak are not strongly identifiable. It is less sensitive to large measurement errors than direct comparison of estimated slope changes.
1
Introduction
This paper proposes a novel approach for detection of dicrotic notches and peaks from
blood volume pulse (BVP) signals. The BVP waveform, which is shown in Fig 1.a, resembles
arterial blood pressure (ABP) waveform. BVP signals, however, are measured by noninvasive photoplethysmography (PPG) sensors, while ABP signals require invasive measurement by a licensed doctor. It has been reported that BVP signals can be successfully used
to derive heart rate signals [1-3]. In particular, dicrotic notches and peaks provide important
information about the cardiovascular system [4],[5]. Their location corresponds to the closure
of the aortic valve and subsequent retrograde blood flow, and can be used to monitor cardiac
function. PPG sensors are usually placed on the patients’ fingers, ears, or toes. The signals
are therefore often distorted by poor periphery perfusion or motion artifacts. The proposed
method is more robust than detection of the largest estimated slope change, as illustrated in
Fig 2 and Fig 3, and overcomes the distortions even for weak dicrotic notches and peaks.
2
Detection Methods
Most dicrotic notch and peak detection algorithms analyze the shape of the waveform (see
Fig 1.a). Kyle et al. [6] first identify the steep upslope of the onset of the cardiac cycle, and
then look for a small upslope in the descending part of the waveform. Kinias et al. [7] iteratively create two chords of decreasing length around a bend-point in the waveform, until its
precise location is found. The dicrotic notch is detected using the slope on either side of the
bend-point. Oppenheim and Sittig [8] improve this method by combining it with analysis of
the signal’s derivatives. Li et al. [9] use inflection and zero-crossing points, and the derivative
of the signal to locate the dicrotic notch.
Our proposed method finds points with the most significant slope changes in the BVP
waveform. This is achieved by comparing the fit of broken-stick and straight-line models in a
sliding time window, as shown in Fig 1. Weighted broken-stick models with different scales
and shapes are used in sequence to detect the following features in the BVP waveform: the
onset, systolic peak, dicrotic notch, and dicrotic peak. See Fig 4 – Fig 7. To improve the reliability of the algorithm, estimates are confirmed using prediction intervals based on previous
cardiac cycles.
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Systolic Peak
Dicrotic Notch
Dicrotic Peak
RSS Ratio
Blood volume pulse signal
Onset
Time
Dicrotic Peak
RSS Ratio
Blood volume pulse signal
(a) Detection of Dicrotic Notch
Time
(b) Detection of Dicrotic Peak
Fig 1. Estimation of the location of a dicrotic notch and a dicrotic peak based on the comparison of two weighted linear models. The locations are estimated at the points of
maxima of the statistics N(t) and P(t). Both statistics are based on the ratio of the sum
of residual errors of the two models. This approach is inspired by hierarchical linear
model comparison theory.
2.1
The Models
Assume that BVP values x1, x2, … are recorded sequentially at times t1 < t2 < …, and consider a sliding time interval represented by a sequence of n-dimensional column vectors t k,n =
(tk+1, tk+2, …, tk+n)ʹ′, k ≥ 0. The corresponding observed BVP signal values will be denoted as
x k,n = (xk+1, …, xk+n)ʹ′. For observations in the k-th time window, we compare the fit of the following two linear models:
(1)
M1: xi = α0 + α1 ti + εi
k+1 ≤ i ≤ k+n
M2: xi = β0 + β1 ti + εi , k+1 ≤ i ≤ k+m
xi = β0 + β1 ti + β2 (ti – T) + εi , k+m < i ≤ k+n,
where εi is the i-th measurement error, α0, α1 and β0, β1, β2 are unknown parameters, and T is
the break point of the broken-stick model M2. In general, T lies somewhere in the interval
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BIOSIGNAL 2010
(tk+m, tk+m+1] for some selected index m with 1 ≤ m ≤ n. If its location is to be estimated, then
the estimation problem becomes non-linear [10-12]. For the sake of simplicity we will therefore use T = T(k,n,m) = (tk+m + tk+m+1) / 2.
The BVP waveform may have time-varying non-linearities near its peaks and notches, that
the broken-stick model M2 is unable to model. They may, however, distort fitting of the
model when its break point is near a peak or notch. It may therefore be useful to exclude or
discount possibly large residual errors near the break point by using zero or near-zero weights
for i close to m, and weights equal to 1 elsewhere. We therefore use weighted least-squares
regression, and view the weights as a part of the design of the algorithm.
2.2
Comparison of Model Fit
The model M1 is a submodel of M2 (with β2 = 0), and the models have p1 = 2 and p2 = 3
parameters, respectively. We will compare the fit of the models using the ratio of their residual sums of squares
(2)
R(k,n,m) = RSS1(k,n,m) / RSS2(k,n,m).
The parameters α0, α1 and β0, β1, β2 are estimated via weighted least-squares regression with
weights wi ≥ 0, and the residual sum of squares for model Mj, with j = 1, 2, is calculated as
n
(3)
RSS j (k,n,m) = ∑ w i (x k +i − xˆ kj+i ) 2 ,
i=1
where xˆ kj+i is the value fitted by model Mj at time tk+i. The statistic R(k,n,m) is related to the
F-statistic
€
(RSS1 − RSS2 ) /( p2 − p1 )
= R(k,n,m) a − a
F=
(4)
RSS2 /(n − p2 )
€
that is used in theory of hierarchical linear models for testing a hypothesis H that the two
models have the same fit [13]. In our application we need to find a time window with the
€ slope change, i.e. with the maximum value of the F-statistic. Since the quanmost significant
tity a = (n–p)/(p2–p1) = n–3 is constant for a fixed window size n, it is enough to compare the
values of R(k,n,m).
The classical test rejects the hypothesis H if the statistic F exceeds a preselected quantile of
its distribution. If the scaled measurement errors w i ε i are i.i.d. standard Gaussian, then the
F-statistic has the F-distribution with p2 – p1 = 1 and n – p2 = n – 3 degrees of freedom
[11],[13]. Note that such assumption about the error distribution requires wi > 0 for all i. The
F-statistics in overlapping windows are not independent, and finding the distribution of their
maximum is a non-trivial task that will €
be addressed elsewhere. For our purpose of finding a
window with the largest F-statistic, i.e. the largest R(k,n,m), we can thus relax the assumptions about the errors distribution. We can then also allow some of the weights wi to be zero
and exclude the possibly large errors near the break point of the broken-stick model.
2.3
Detection of Slope Changes
^
If there is an estimated slope increase in the broken-stick model M2 (i.e. if β2 > 0) then we
suspect a possible notch at the break point, and use R(k,n,m) as a notch detection statistic
N(k,n,m). Otherwise R(k,n,m) will represent a statistic P(k,n,m) for peak detection. Notice that
R(k,n,m) ≥ 1 because M1 is a submodel of M2. We can thus define the detection statistics as
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(5)
RSS Ratio
Blood volume pulse
The statistics R, N, and P can be maximized in a given time interval to detect the most significant slope change, increase (notch), or decrease (peak), respectively. Our proposed algorithm uses these statistics in either forward or backward mode in a given search interval to
find either the first or last maximum above a given threshold. Fig 1 illustrates this method for
detection of dicrotic notch and peak. The search interval is determined in multi-resolution
manner as described in Tab 1, and confirmed using a prediction interval that is based on previous dicrotic notch and peak estimates.
Time
RSS Ratio
Blood volume pulse
(a) Original Data: Both Models Fit
(b) Data with Outlier: Neither Model Fits
Time
Fig 2. Even though the effects of an outlier may be very strong when estimating slope
changes, the slope change in (b) is not significant since the broken-stick model does
not fit the data better than the straight-line. The statistic N(t) is similar in (a), (b), and
thus robust. The corresponding detection statistics are depicted in Fig 3.
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RSS Ratio N(t)
Slope Change
2.4
Robustness of the Detection Statistics
One might argue that the statistics proposed in (5) are too complicated for the purpose of
finding the bend-points of the cardiac waveform. Indeed, it seems more natural to simply
consider points where the estimated slope change is the largest. Such an intuitive approach,
however, cannot measure whether the estimated slope change is real, or if it is caused by
measurement errors or outliers.
The main advantage of the proposed statistics (5) is that, while being sensitive to real slope
changes, they tend to ignore erroneous estimated slope changes. This is achieved by finding
the most significant slope change, not the largest slope change. Comparing the fit of two
models M1 and M2 is essential for this task. The robustness of the procedure is illustrated in
Fig 2 and Fig 3 by inserting an artificial outlier into the BVP waveform. The algorithm views
a small real slope change as more significant than an estimated large slope change that was
caused by measurement errors. As result, the proposed algorithm is robust with respect to
measurement errors, outliers, and signal distortions.
Detail below
Time
Time
(b) Largest slope change
RSS Ratio N(t)
(a) Most significant slope change
Time
(c) Detail from Fig (a)
Fig 3. The effects of the outlier from Fig 2 on the detection statistics. The proposed detection
statistic N(t) is almost unaffected as shown in Fig (a) and (c). The outlier has drastic
effects on a detection statistic that is based on the magnitude of the estimated slope
change in Fig (b) (solid red line). The effect in (b) is in fact so strong, that it could
affect the location of the maximum of the statistic, and lead to erroneous dicrotic
notch detection at the location of the outlier.
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3
The Detection Algorithm
Highest
Resolution
High Resolution
Analysis
Low Resolution Analysis
The main steps of the multi-resolution detection algorithm are summarized in Tab 1 and illustrated by Fig 1 and Fig 4 – Fig 7.
(1) Detect the onset of the systolic upstroke
Detect two consecutive large-scale notches using statistic N(k,n1,m1) for k starting
after the latest detected dicrotic peak. These two notches serve as an estimate of
the current cardiac cycle.
(2) Detect the systolic peak
Detect the first large-scale peak using statistic P(k,n2,m2) for k in the current cardiac cycle. Confirm that the peak falls into a prediction interval.
(3) Detect the dicrotic notch
Detect a small-scale notch using statistic N(k,n3,m3) for k in a prediction interval
that is based on previous cycle estimates.
(4) Detect the dicrotic peak
Detect a small-scale peak using statistic P(k,n4,m4) for k in a prediction interval
that is based on previous cycle estimates.
(5) Detect the dicrotic notch at smaller scale if necessary
If the dicrotic notch was missed in step (3), detect the notch backwards from the
peak detected in (4), using smaller scale models than in (3), e.g. n5 =n3/2.
(6) Return to step (1) to estimate the next cardiac cycle bounds.
Tab 1. Multi-resolution algorithm for dicrotic notch and peak detection.
4
Results of Experimental Evaluation
Our proposed algorithm is related to detecting bend-points in the waveform, but it has the
following main advantages: 1. It is very robust, as illustrated by Fig 2, because it measures
the fit improvement as opposed to angle change. 2. It considers bend-points at different scales
to detect different features of the waveform. 3. It uses weighted regression to discount nonlinearities at the bend-points.
Experimental evaluation of the algorithm was performed using BVP signals that were collected from student volunteers at National Cheng Kung University, Taiwan. The algorithm
performed well even for distorted signals with weak dicrotic notch and peak (Fig 8). We have
tested the performance of the algorithm using 1355 cardiac cycles where the presence and
locations of dicrotic notches and peaks were determined by manual analysis. The error rates
of the algorithm are listed in Tab 2. In all situations, the rates are approximately 4% or less,
for both missed and false positive detections. The accuracy of valid detections is summarized
in Tab 3. The average absolute error is approximately 7.5 msec for both dicrotic notch and
peak detection. An interesting observation is that the bias of the estimates is close to half of
the average absolute error, and that it has different sign for notches and the peaks. These facts
make us believe that the accuracy of the algorithm can be further improved by reducing its
bias. Bias reduction of the algorithm will be addressed elsewhere in the future.
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BVP Signal
RSS Ratio
BIOSIGNAL 2010
BVP Signal
RSS Ratio
Time
Time
BVP Signal
RSS Ratio
Fig 4. Large-scale detection of the onset to determine the cardiac cycle boundaries
Time
Fig 5. The large-scale notch detection statistic N(t) from Fig 4 is insensitive to small-scale
notches. Dicrotic notches are detected on a smaller scale as shown in Fig 1.a.
RSS Ratio
BVP Signal
Time
Fig 6. Large-scale detection of the systolic peak.
RSS Ratio
BVP Signal
Time
Fig 7. The large-scale peak detection statistic P(t) from Fig 6 is insensitive to small-scale
peaks. Dicrotic peaks are detected on a smaller scale as shown in Fig 1.b.
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Blood volume pulse
BIOSIGNAL 2010
Time
Fig 8. Experiments – the algorithm safely recovers from the missing dicrotic peak at t = 411
and detects dicrotic notches and peaks that are not easily identifiable.
Feature
Dicrotic notch
Dicrotic peak
Missed Detections
44/1355 = 0.032
58/1355 = 0.043
False Detections
26/1337 = 0.019
55/1352 = 0.040
Tab 2. Number of features that were missed or incorrectly identified by the proposed algorithm. The denominators in the respective columns correspond to: total number of
features identified by a human, total number of features detected by the algorithm; total number of valid detections.
Feature
Dicrotic notch
Dicrotic peak
Mean Absolute
Error (msec)
7.6
7.4
Bias
(msec)
4.3
-3.2
Standard
Deviation
8.13
8.38
Standard
Error
0.22
0.23
Tab 3. The accuracy of valid detections of the proposed algorithm, compared with human
identified features. The bias corresponds to approximately half of the mean absolute
error. Reducing the bias has the potential to improve the accuracy of the algorithm.
5
Conclusions
The PPG waveform is similar to ABP waveforms that is important for monitoring cardiac
function, but PPG sensors are noninvasive while ABP requires invasive measurement by a
licensed doctor. The PPG sensors have several other advantages, including low cost, ease of
use, widespread availability, and accessibility to various sites of human body. On the other
hand, the PGP signals can easily be distorted. The main advantage of the proposed dicrotic
notch and peak detection procedure is that it is robust with respect to measurement errors and
outliers as discussed in Section 2.4 and illustrated in Fig 2 and Fig 3. The proposed algorithm
thus shows great promise in dealing with the distortions of the BVP signals, and has the potential to improve BVP-based cardiac function monitoring.
Acknowledgement
This work has been partly supported by grant project NSC 97-3114-E-006-001, National
Science Council, Taiwan.
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References
[1] Murthy VS, Ramamoorthy S, Srinivasan N, Rajagopal S, Rao MJ. Analysis of Photoplethysmographic Signals of Cardiovascular Patients. 1998; .
[2] Gu YY, Zhang YT. Reducing the influence of contacting force applied on photoplethysmographic sensor on heart rate variability estimation. Engineering in Medicine
and Biology Society, 2003. Proceedings of the 25th Annual International Conference of
the IEEE 2003; 3.
[3] Lu S, Zhao H, Ju K, Shin K, Lee M, Shelley K, Chon KH. Can Photoplethysmography
Variability Serve as an Alternative Approach to Obtain Heart Rate Variability Information? Journal of clinical monitoring and computing 2008; 22(1):23-29.
[4] Haluska BA, Jeffriess L, Mottram PM, Carlier SG, Marwick TH. A new technique for
assessing arterial pressure wave forms and central pressure with tissue Doppler. Cardiovascular Ultrasound 5:6-6.
[5] Sesso HD, Stampfer MJ, Rosner B, Hennekens CH, Gaziano JM, Manson JE, Glynn RJ.
Systolic and diastolic blood pressure, pulse pressure, and mean arterial pressure as predictors of cardiovascular disease risk in Men. Hypertension 2000; 36(5):801-807.
[6] Kyle MC, Klingeman JD, Freis ED. Computer identification of brachial arterial pulse
waves. Computers and Biomedical Research 1968; 2(2):151-159.
[7] Kinias P, Fozzard H, Norusis M. A real-time pressure algorithm. Computers in Biology
and Medicine 1981; 11(4):211-220.
[8] Oppenheim MJ, Sittig DF. An Innovative Dicrotic Notch Detection Algorithm Which
Combines Rule-Based Logic with Digital Signal Processing Techniques. Computers and
Biomedical Research 1995; 28(2):154-170.
[9] Li BN, DONG MC, Vai MI. On an automatic delineator for arterial blood pressure
waveforms. Biomedical Signal Processing and Control 2010; 5(1):76-81.
[10] Farley J, Hinich M. A test for a shifting slope coefficient in a linear model. Journal of
the American Statistical Association 1970; 65(331):1320-1329.
[11] Hinkley D. Inference about the intersection in two-phase regression. Biometrika 1969;
56(3):495-504.
[12] Bacon DW, Watts DG. Estimating the transition between two intersecting straight lines.
Biometrika 1971; 58(3):525-534.
[13] Seber G, Lee A. Linear regression analysis. New York: Wiley, 1977.
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